Mathematical Models: Probabilistic

Mathematical Models: Probabilistic

Introduction

In the first basic statistics course, empirical distributions were studied, first it was presented geometrically and then with a partial arithmetic representation, through average and standard deviation. If when studying the behavior of a random variable it is seen that it behaves in a certain way, it is possible to use known models to calculate the probability of an event occurring. That is, a probabilistic model that allows describing the results of an experiment, as well as predicting the behavior of the study variable. 

Frequently, probabilistic models are also called probability distributions. It is important to note that there are probabilistic models for both discrete and continuous variables. This course will study two of the discrete distributions: binomial and poisson;But there are more, such as geometric and hypergeometric distribution, while, of continuous distributions, in the first unit we will only see the normal distribution.

Binomial

A distribution is considered binomial when:

• The events that are presented are independent.
• There are only two possible results of the event.
• The probability of success remains constant.
• The random variable X is defined as the number of successes within a fixed number of trials.

If p is the probability of success, q 1 p is the probability of failure, x is the specific number of successes and n the number of trials, then the probability p that x successes occur in n tests is: 

x n x p x ncx p q Another way of writing probability:

x n x p q x n x n p x n p ! For this distribution, you have to:

• • The average: NP
• • The variance: NPQ 2 
• • The standard distribution: NPQ

Poisson

This distribution is used to calculate the probability of a designated number of events when:

• Events occur in a time or space continuum.
• Events occur independently.
• The events are "rare" p 0.1 and NP 5.

Theoretically, the possibilities in this type of distribution are endless;that is to say that the number of events goes from zero to infinity discreetly. To determine the probability that a certain number of successes occur in a Poisson process, it is only necessary to know the average, long -term number of events for time or space of interest, said average value is designated. One of the care that must be taken when using the Poisson distribution formula is that the value of  must be applied to the relevant period of time. The probability of x successes in a Poisson distribution is given by: x e

Normal

This probabilities distribution is continuous and symmetrical, that is, with the observed values distributed uniformly and also, it is not flat or pointed mecritic pointed. The normal distribution is important for three reasons:

• • Many random processes behave in this way.
• • They are used to approximate other probability distributions, such as binomial and Poisson.
• • The probability distribution of the sample mean and the sample proportion is the normal distribution when the sample size is large, regardless of the form of the population of origin.

In the case of a random variable with continuous probability distribution, it is only possible to determine the probability value that the random variable takes values in an interval;Since there is an infinite number of values in any interval, the probability that it takes a particular value is zero

If the limit form of a histogram for a frequency distribution has the shape of a bell, then a normal curve can be used for the determination of probabilities. Remember that for a continuous variable it is not possible to know the probability of an event, so it is necessary to make frequency distributions. It is known that  has the following geometric interpretation with respect to the normal curve.

• • The area under the normal curve between and is approximately 68% of the total area.
• • The area under the normal curve between 2 and 2 is approximately 95% of the total area.
• • The area under the normal curve between 3 and 3 is approximately 99.7% of the total area.

Problem

Assume that a certain eye color feature, being left -handed, etc. It is determined by a couple of genes, and which also represents a dominant gene, and r a recessive gene. A public security officer with a couple of genes D, d is said to be pure dominant and with the gene couple R, r it is said that it is pure recessive and with a couple d, r is said to be hybrid. Apparently, pure dominant and hybrids are similar. The descendants of a couple receive a gene from each parent and this gene can, with the same probability, be one of the two that the aforementioned parent has. 

conclusion

We recover here the information presented in the first unit of the course referring to statistical models. When the researcher poses an experimental design and begins with data collection, it is because he pursues the verification study of an objective on the population under study. 

These objectives are usually established based on theories or hypotheses that wish to verify on the functioning of the population under certain experimental conditions. For example: theories that establish the possible relationship between two characteristics of the population, theories that put the idea of different behaviors for a characteristic of the population based on a variable that classifies the subjects under study in different groups.

A first step in statistical modeling is the approach of a mathematical expression that represents the general behavior of the population under study taking into account the established experimental design and the objective or objectives that are desired to verify. This is what is known as the systematic component of the model and focuses solely on the controlling part of our experimental design. 

For example, if we consider knowing the sum of two numbers everyone knows what is the mathematical function that allows the sum of unique way. This is called a deterministic function because it always provides the same result if the input values are equal.